On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients

Abstract

Abstract. In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the function that appears in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in return to find identities involving binary quadratic forms.

Binary quadratic forms and the Fourier coefficients of certain weight 1 eta -quotients

“…Define S 1 , S 2 , S 3 , S 4 to be the set of primes p = 71 represented by (1, 1, 18), (2,1,9), (4, 3, 5), and (3, 1, 6), respectively. Let S 5 be the set of primes p with −71 p = −1.…”

“…The general formula is more involved and will be discussed elsewhere. Principal Genus (1, 1, 34), (4,3,9), (4, −3, 9) +1 +1 Second Genus (5,5,8), (2,1,17), (2, −1, 17) −1 −1 . In the above table, p is taken to be coprime to −135 and represented by the given genus.…”

“…(3) p is represented by the form (5,5,8) if and only if W −135 (x) factors into three irreducible quadratic polynomials modulo p, (4) p is represented by the form (2,1,17) if and only if W −135 (x) remains irreducible modulo p.…”

“…It is a continuing area of research to determine the Fourier coefficients of certain eta-quotients. (See [1], [2], [6], [7], [13], and [22]. )…”

“…We would like to point out that many special cases of Theorem 2.1 were previously discussed in the literature. (See [2], [4], [10], [14], [15], [16], [18], [20]. )…”